Aidy Boothroyd's side are still not mathematically safe from relegation after their game in hand was all over by the break.
Goals from Neilson, Clarke and a Kitson own goal pushed Crawley back into the promotion places, with a consolation strike from Brett Williams all Northampton had to show for their efforts.
The Cobblers now go to Hereford United on Saturday looking to secure their League Two status and stop a promising run of form under Boothroyd tailing off even further.
“It was a really strange game and I can’t remember a half of football like that first one,” said the manager afterwards.
“If you take the goals out of it you’d probably say it was a decent half, but we conceded three really poor goals, all from defensive errors.
“They could all have been avoided, and should have been. We hit the crossbar, got a few strange decisions and I just told the players to go out and win the second half.
“We knew if we got one they would start to twitch a little bit because of the position they are in the league and we hit the woodwork twice after that.
“It was a bit like the Bradford game where we had the opportunities but we conceded three daft goals.”